Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | Oct-Nov | (P1-9709/01) | Q#8

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Question

A function f is defined by  for .

i.       Find an expression, in terms of , for  and show that f is an increasing function.

ii.       Find an expression, in terms of , for  and find the domain of .

Solution

i.

We have the function;

The expression for  represents derivative of .

Rule for differentiation is of  is:

Rule for differentiation is of  is:

Rule for differentiation is of  is:

To test whether a function  is increasing or decreasing at a particular point , we  take derivative of a function at that point.

If  , the function  is increasing.

If  , the function  is decreasing.

If  , the test is inconclusive.

Since  is positive for all values of ,  will be positive, always.

Hence  is an increasing function.

ii.

We have;

We write it as;

To find the inverse of a given function  we need to write it in terms of  rather than in terms of .

Interchanging ‘x’ with ‘y’;

The set of numbers for which function  is defined is called domain of the function.

Finding domain of a function :

·       Since denominator of a fraction cannot be zero, equate denominator of function  with zero and solve for set of values of  which produce zero in the denominator. Exclude these values  of  from domain of

·       Since the number under square root must be positive , make the inequality of number under  square root with  and solve for set of values of  which produce number under square root.  Exclude these values of  from domain of

At a glance, we might be tempted to find domain of  by considering . That is an  option if  were defined for all .

Domain and range of a function  become range and domain, respectively, of its inverse function .

Domain of a function  Range of

Range of a function  Domain of

Therefore, if we have range of  , we can find the domain of .

The function f is defined by  for . Therefore, we can find the range of   by substituting the extreme values of  in it.

At ;

At ;

Hence range of ;

Since range of a function is domain of its inverse function, the domain of  is;